Question

What is the maximum number of electrons in an atom that can have the following quantum numbers? (a) $n=3,m_l=-1;$ (b) $n=4,l=2$; (c) $n=4,l=3,m_l=-2;$ (d) $n=5,l=2$, $m_l=0.$

Short answer

(a) For the quantum numbers n=3, m_l=-1, the maximum number of electrons in an atom is 2. (b) For the quantum numbers n=4 and l=2, the maximum number of electrons in an atom is 10. (c) For the quantum numbers n=4, l=3, and m_l=-2, the maximum number of electrons in an atom is 2. (d) For the quantum numbers n=5, l=2, and m_l=0, the maximum number of electrons in an atom is 2.

Step-by-step solution

Understanding quantum numbers

The principal quantum number (n) determines the energy level of an electron in an atom. It can have positive integer values (n = 1, 2, 3,...). The greater the n value, the higher the energy level. The orbital angular momentum quantum number (l) defines the shape of an electron's orbital. It depends on the value of n, and can take integer values from 0 to (n-1). For each specific value of n, there are n-l orbitals, and different shapes. The magnetic quantum number (m_l) describes the orientation of the electron's orbitals in space. It depends on the value of l and can take integer values from -l to +l, including zero. Thus, for a given l value, there are 2l+1 possible m_l values.

(a) Quantum numbers: n=3, m_l=-1

First, let's determine the possible values of l for the given n value. Since n=3, l can take the values 0, 1, and 2. Now let's find the possible values of m_l for each l value: - For l=0, there is only one possible m_l value: 0 - For l=1, m_l can have the values -1, 0, and +1 - For l=2, m_l can have the values -2, -1, 0, +1, and +2 The given m_l value is -1, which corresponds to l=1 and l=2. However, we need to find the maximum number of electrons for a set of quantum numbers. As l=2 can hold more electrons than l=1, we'll consider l=2 for this case. The number of electrons in an orbital can have either of two possible spin quantum numbers, +1/2 (spin up) or -1/2 (spin down). Therefore, an orbital with given n, l, and m_l values can hold a maximum of 2 electrons.

Answer: (a)

For the quantum numbers n=3, m_l=-1, the maximum number of electrons in an atom is 2.

(b) Quantum numbers: n=4, l=2

For the given n and l values, let's find the possible values of m_l. For l=2, m_l can have the values: -2, -1, 0, +1, and +2. Since each orbital can hold a maximum of 2 electrons, we can calculate the total number of electrons for the given n and l values by multiplying the number of m_l values by 2: Number of electrons = 2 * (Number of m_l values) = 2 * 5 = 10

Answer: (b)

For the quantum numbers n=4 and l=2, the maximum number of electrons in an atom is 10.

(c) Quantum numbers: n=4, l=3, m_l=-2

Since all quantum numbers are given in this case, we just have to consider the number of electrons that can exist in a single orbital for the given quantum numbers. As mentioned before, a single orbital can hold a maximum of 2 electrons.

Answer: (c)

For the quantum numbers n=4, l=3, and m_l=-2, the maximum number of electrons in an atom is 2.

(d) Quantum numbers: n=5, l=2, m_l=0

Like in the case (c), all quantum numbers are given, so we only need to consider the number of electrons that can exist in a single orbital for the given quantum numbers. A single orbital can hold a maximum of 2 electrons.

Answer: (d)

For the quantum numbers n=5, l=2, and m_l=0, the maximum number of electrons in an atom is 2.

Key concepts

principal quantum number

The principal quantum number, denoted by **n**, is an essential component of quantum mechanics. It primarily dictates the main energy level or shell in which an electron resides within an atom. The principal quantum number can be any positive integer (1, 2, 3, ...). The higher the value of **n**, the larger the size of the orbital and the greater the energy level of the electron. This means that electrons in higher energy levels are farther from the nucleus.

Understanding the significance of the principal quantum number is crucial, as it lays the foundation for determining the possible values of the other quantum numbers. For instance:

• When **n=1**, the first energy level (also known as the ground state) can only accommodate **s-orbitals** (which we denote by **l = 0**).
• As **n** increases, more complex orbitals (such as p, d, and f) can be present due to an increased range of **l** values.

Thus, knowing the principal quantum number helps you understand not only the energy level but also the potential complexity and diversity of electron orbitals within that level.

orbital angular momentum

The orbital angular momentum quantum number, symbolized by **l**, defines the shape and type of the electron's orbital within a given energy level. The value of **l** is determined by the principal quantum number **n**, and can range from 0 to **n-1**. Each value of **l** corresponds to a specific type of orbital with its characteristic shape.

Here is a breakdown:

• For **l=0**, we have **s-orbitals**, which are spherical.
• For **l=1**, we have **p-orbitals**, which are dumbbell-shaped.
• For **l=2**, we encounter **d-orbitals**, more complex and often cloverleaf-shaped.
• For **l=3**, the orbital type is **f**, with even more intricate shapes.

These shapes are important not just for their theoretical interest, but also for practical applications, such as predicting how electrons are distributed in atoms, how they interact with each other, and how they bond to form molecules.

Knowing the orbital angular momentum helps in visualizing the spatial distribution of electrons, which is vital for chemical bonding and reactions.

magnetic quantum number

The magnetic quantum number, labeled **m_l**, specifies the orientation of an orbital in space within a magnetic field. It directly depends on the values of the orbital angular momentum quantum number **l**. The possible values of **m_l** range from -l to +l, including zero. Consequently, for any given **l**, there are usually 2l+1 possible orientations for an orbital.

An understanding of the magnetic quantum number allows us to ascertain how many distinct orbitals are available within a given sublevel. For example:

• When **l=0**, the value of **m_l** is solely 0. Hence, only one orientation (or 1 orbital) is possible.
• For **l=1**, **m_l** covers values -1, 0, and +1, resulting in 3 possible orientations.
• When **l=2**, **m_l** values include -2, -1, 0, +1, and +2, leading to 5 distinct orientations.

The role of the magnetic quantum number becomes prominent in more advanced settings. For instance, it assists in understanding the fine structure of electron spectra or in interpreting how atoms behave under magnetic fields.

electron configuration

Electron configuration refers to the distribution of electrons among the different orbitals or energy levels of an atom. This concept is fundamental because it determines the chemical behavior and properties of an element. The way electrons fill up or "occupy" available orbitals follows specific principles, notably the Pauli exclusion principle and Hund’s rule.

• The **Pauli exclusion principle** states that no two electrons in an atom can have identical quantum numbers. This ensures that each orbital can hold a maximum of two electrons, which must have opposing spins.
• **Hund’s rule** emphasizes that electrons will fill up empty orbitals of the same energy level before pairing up in a single orbital.

For any given atom, the sequence of filling up these levels and sublevels can be denoted using standard notation, such as **1s² 2s² 2p⁶**, clearly indicating which orbitals are filled and how many electrons are in each.

Properly understanding electron configuration is essential, not just for chemistry homework but also for grasping key concepts in chemical bonding, reactivity, and molecular structure.